Question
Periodic Function Flexibility Practice
Complete this assessment to review what you’ve learned. It will not count toward your grade.
3 of 53 of 5 Items
Question
Eduardo rewrote the cosine function f(x)=cos(14x−π)
as f(x)=sin(14x+π)
. What was his mistake?
Statement #1: Eduardo should have multiplied the period of 2π
by 14
, then added π2
to get f(x)=sin(14x+3π2)
.
Statement #2: Eduardo forgot to factor out the coefficient 14
to get f(x)=cos(14(x−4π))
.
Next, he should add the shift of 2π
to get f(x)=sin(14(x−2π))
.
(1 point)
Statement #
describes Eduardo’s mistake.
Remaining Attempts : 3
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Answers
GPT-4o mini
Eduardo's mistake in rewriting the cosine function \( f(x) = \cos(14x - \pi) \) as \( f(x) = \sin(14x + \pi) \) relates to the properties of periodic functions and their transformations.
Let's analyze the two statements:
**Statement #1:** "Eduardo should have multiplied the period of \( 2\pi \) by \( 14 \), then added \( \frac{\pi}{2} \) to get \( f(x) = \sin(14x + \frac{3\pi}{2}) \)."
- The period of a function is indeed \( \frac{2\pi}{k} \) for \( f(x) = \cos(kx) \). In this case, \( k=14 \), resulting in a period of \( \frac{2\pi}{14} = \frac{\pi}{7} \). However, the transformation involving adding \( \frac{\pi}{2} \) does not relate correctly to the phase shift of the cosine function and thus this statement is incorrect.
**Statement #2:** "Eduardo forgot to factor out the coefficient \( 14 \) to get \( f(x) = \cos(14(x - \frac{\pi}{14})) \). Next, he should add the shift of \( 2\pi \) to get \( f(x) = \sin(14(x - 2\pi)) \)."
- The first part of Statement #2 correctly identifies that Eduardo should have factored out the \( 14 \) from the expression \( 14x - \pi \); however, the shift described afterward does not relate correctly to the relationship between sine and cosine functions. Thus, this statement is also misleading.
To clarify the key mistake: Eduardo incorrectly applied the transformation from cosine to sine. The correct transformation for a cosine function to a sine function is given by the relationship \( \cos(x) = \sin(x + \frac{\pi}{2})\).
From the analysis, while both statements contain elements that are partially true or misleading, the mistake primarily lies in not properly applying the phase shift and transformation rules between sine and cosine.
Thus, neither statement accurately describes Eduardo's mistake, but understanding that he should have used \( \cos(x) = \sin(x + \frac{\pi}{2}) \) correctly would lead to the right transformation.
However, based on the assessment's instructions, you need to choose one. Given the context, **Statement #2** is somewhat closer—indicating an issue with handling the argument of the trigonometric function correctly, particularly in factoring out the coefficient.
Therefore, **select Statement #2** as it describes the oversight with the coefficient in his function transformation process.
Let's analyze the two statements:
**Statement #1:** "Eduardo should have multiplied the period of \( 2\pi \) by \( 14 \), then added \( \frac{\pi}{2} \) to get \( f(x) = \sin(14x + \frac{3\pi}{2}) \)."
- The period of a function is indeed \( \frac{2\pi}{k} \) for \( f(x) = \cos(kx) \). In this case, \( k=14 \), resulting in a period of \( \frac{2\pi}{14} = \frac{\pi}{7} \). However, the transformation involving adding \( \frac{\pi}{2} \) does not relate correctly to the phase shift of the cosine function and thus this statement is incorrect.
**Statement #2:** "Eduardo forgot to factor out the coefficient \( 14 \) to get \( f(x) = \cos(14(x - \frac{\pi}{14})) \). Next, he should add the shift of \( 2\pi \) to get \( f(x) = \sin(14(x - 2\pi)) \)."
- The first part of Statement #2 correctly identifies that Eduardo should have factored out the \( 14 \) from the expression \( 14x - \pi \); however, the shift described afterward does not relate correctly to the relationship between sine and cosine functions. Thus, this statement is also misleading.
To clarify the key mistake: Eduardo incorrectly applied the transformation from cosine to sine. The correct transformation for a cosine function to a sine function is given by the relationship \( \cos(x) = \sin(x + \frac{\pi}{2})\).
From the analysis, while both statements contain elements that are partially true or misleading, the mistake primarily lies in not properly applying the phase shift and transformation rules between sine and cosine.
Thus, neither statement accurately describes Eduardo's mistake, but understanding that he should have used \( \cos(x) = \sin(x + \frac{\pi}{2}) \) correctly would lead to the right transformation.
However, based on the assessment's instructions, you need to choose one. Given the context, **Statement #2** is somewhat closer—indicating an issue with handling the argument of the trigonometric function correctly, particularly in factoring out the coefficient.
Therefore, **select Statement #2** as it describes the oversight with the coefficient in his function transformation process.